Harry has three coins.
- One coin is biased so that, when it is thrown, the probability of obtaining a head is \(\frac{1}{3}\).
- The second coin is biased so that, when it is thrown, the probability of obtaining a head is \(\frac{1}{4}\).
- The third coin is biased so that, when it is thrown, the probability of obtaining a head is \(\frac{1}{5}\).

The random variable \(X\) is the number of heads that Harry obtains when he throws all three coins together.
(a) Find the probability generating function of \(X\).

Isaac has two fair coins. The random variable \(Y\) is the number of heads that Isaac obtains when he throws both of his coins together. The random variable \(Z\) is the total number of heads obtained when Harry throws his three coins and Isaac throws his two coins.
(b) Find the probability generating function of \(Z\), expressing your answer as a polynomial in \(t\).
(c) Use the probability generating function of \(Z\) to find \(\mathrm{E}(Z)\).